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<h1>Wavelet transform mass detector</h1>

<h2>Description</h2>

<p>
The Wavelet transform mass detector is particularly suitable for low-resolution and noisy data.
The method uses the Mexican Hat wavelet model of the continuous wavelet transform CWT) algorithm. 
The search of mass spectrum peaks is executed in three steps. 
First, the data point intesity is converted into wavelet domain.
Next, all local maxima of the calculated wavelet are found. 
Finally, m/z peaks (ions) are declared in those points, where the wavelet has a local maximum.  
The m/z peak is formed with the selected data point (mass and intensity) using the wavelet and all surrounding data points. 
The final m/z value of the ion is calculated as an average of m/z values of surrounding data points weighted by their intensity.
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<h4>Mathematical model</h4>

<p>
In mathematics and numerical analysis, the Mexican hat wavelet is the normalized second derivative of a Gaussian function.<br>
<img src="WaveletFormula.png"><br>
The parameter <i>t</i> is the intensity of each data point in the curve, and <i>sigma</i> corresponds to the standard deviation.
</p>

<p>
<img src="Wavelet_Mex_Hat.png">
<img src="Wavelet_calculated_form.jpg">
</p>

<p>
To simplify the process of wavelet calculation, the original function 
is transformed into two parts, where <i>Wc</i> is the wavelet coefficient and <i>y</i>
is the intensity of the wavelet at certain point. In the following formula, "t" is the
<i>Wavelet window size(%)</i> parameter.
</p>

<p>
<img src="Wavelet_coefficients_calculation.jpg">
<img src="Wavelet_intensities_calculation.jpg">
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<p>
The limits, where the Mexican Hat wavelet is evaluated, are from -5 until 5 (ESL, ESR)
 and the incremental step used in this range is the result of divide the width of ESL 
 to ESR range by 60,000. The number of coefficients used to calculated the wavelet intensity 
 depends on the <i>Scale level</i> parameter.
</p>

<h4>Method parameters</h4>
<dl>
<dt>Noise level</dt>
<dd>The minimum intensity level for a data point to be considered part of a chromatogram. All data points below this intensity level are ignored.</dd>

<dt>Scale level</dt>
<dd>This value is the scale factor that either dilates or compresses the wavelet signal. 
When the scale factor is relatively low, the signal is more contracted which in turn 
results in a more detailed resulting graph and as a consequence more noisy peaks are detected.
 On the other hand, when the scale factor is high, the signal is stretched out which means that
 the resulting graph will be presented in less detail resulting in a smoothed signal.</dd>
 
<dt>Wavelet window size (%)</dt>
<dd>This value is the size of the window used to calculated the wavelet signal. 
When the size of the window is small, more noisy peaks can be detected.
 The proper setting of this parameter may help to avoid the undesired noise
 peaks.<br><br>
 
Wavelet window size at 10%:<br>
<img src="Wavelet_window_size10.jpg"><br>

Wavelet window size at 100%:<br>
<img src="Wavelet_window_size100.jpg"><br>
</dd>

</dl>

<p>
<img src="Wavelet_mass_detector_preview.jpg">
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